Average Value Integral Calculator for C

What is Average Value Integral?

The average value of a function f(x) over an interval [a, b] represents the mean value of the function's outputs over that interval. It's calculated by dividing the integral of the function by the length of the interval.

This concept is particularly useful in physics, engineering, and economics where you need to find the mean behavior of a quantity over a period.

How to Calculate Average Value Integral

To calculate the average value of an integral:

1. Identify the function f(x) and the interval [a, b]
2. Compute the definite integral of f(x) from a to b
3. Divide the result by the length of the interval (b - a)

The result is the average value of the function over the specified interval.

Average Value Integral Formula

Where:

• f(x) is the function
• [a, b] is the interval
• ∫[a to b] f(x) dx is the definite integral of f(x) from a to b

Example Calculation

Let's find the average value of f(x) = x² from x = 0 to x = 2.

1. Compute the integral: ∫[0 to 2] x² dx = (x³/3) evaluated from 0 to 2 = (8/3) - 0 = 8/3
2. Divide by the interval length: (8/3) / (2 - 0) = 8/6 = 4/3 ≈ 1.333

The average value of x² over [0, 2] is 4/3.

Implementing in C

Here's a C program to calculate the average value of an integral:

#include <stdio.h>
#include <math.h>

double integrate(double (*f)(double), double a, double b, int n) {
    double h = (b - a) / n;
    double sum = 0.0;
    for (int i = 0; i < n; i++) {
        double x = a + i * h;
        sum += f(x) * h;
    }
    return sum;
}

double average_value(double (*f)(double), double a, double b, int n) {
    double integral = integrate(f, a, b, n);
    return integral / (b - a);
}

double example_function(double x) {
    return x * x;
}

int main() {
    double a = 0.0, b = 2.0;
    int n = 1000; // Number of rectangles for numerical integration
    double avg = average_value(example_function, a, b, n);
    printf("Average value: %.4f\n", avg);
    return 0;
}

This program uses numerical integration (rectangle method) to approximate the integral, then calculates the average value by dividing by the interval length.